Colloquium article in Reviews of Modern Physics 75 , 913 (2003)

Quantum phase transitions: from Mott insulators to the cuprate superconductors

Colloquium article in Reviews of Modern Physics 75, 913 (2003)

Leon Balents (UCSB) Eugene Demler (Harvard) Matthew Fisher (UCSB) Kwon Park (Maryland) Anatoli Polkovnikov (Harvard) T. Senthil (MIT) Ashvin Vishwanath (MIT) Matthias Vojta (Karlsruhe) Ying Zhang (Maryland)

Parent compound of the high temperature superconductors: 2 4La CuO

Cu

O

La

Band theory

k

k

Half-filled band of Cu 3d orbitals –ground state is predicted by band theory to be a metal.

However, La2CuO4 is a very good insulator

A Mott insulator


Ground state has long-range spin density wave (Néel) order at wavevector K= ()

0 spin density wave order parameter:

; 1 on two sublatticesii i

S

S

spin operator with

angular momentum =1/2

ij i jij

i

H J S S

S

S

A Mott insulator





S

S

spin operator with


ij i jij

i

H J S S

S

S

A Mott insulator





S

S

spin operator with


ij i jij

i

H J S S

S

S

BCS

Doped state is a paramagnet with 0

and also a high temperature superconductor with

the BCS pairing order parameter 0

With increasing , there must be one or more

qua

.

nt

BCS

um phase transitions involving

( ) onset of a non-zero

( ) restoration of spin rotation invariance by a transition

from 0 to 0

i

ii

Introduce mobile carriers of density by substitutional doping of out-of-

plane ions e.g. 2 4La Sr CuO

Superconductivity in a doped Mott insulator

First study magnetic transition in Mott insulators………….

OutlineOutline

A. Magnetic quantum phase transitions in “dimerized” Mott insulators

Landau-Ginzburg-Wilson (LGW) theory

B. Mott insulators with spin S=1/2 per unit cellBerry phases, bond order, and the

breakdown of the LGW paradigm

C. Cuprate SuperconductorsCompeting orders and recent experiments

A. Magnetic quantum phase tranitions in “dimerized” Mott insulators:

Landau-Ginzburg-Wilson (LGW) theory:

Second-order phase transitions described by fluctuations of an order parameter associated with a broken symmetry

TlCuCl3

M. Matsumoto, B. Normand, T.M. Rice, and M. Sigrist, cond-mat/0309440.

S=1/2 spins on coupled dimers

jiij

ij SSJH

10

JJ

Coupled Dimer AntiferromagnetM. P. Gelfand, R. R. P. Singh, and D. A. Huse, Phys. Rev. B 40, 10801-10809 (1989).N. Katoh and M. Imada, J. Phys. Soc. Jpn. 63, 4529 (1994).J. Tworzydlo, O. Y. Osman, C. N. A. van Duin, J. Zaanen, Phys. Rev. B 59, 115 (1999).M. Matsumoto, C. Yasuda, S. Todo, and H. Takayama, Phys. Rev. B 65, 014407 (2002).

Weakly coupled dimersclose to 0

close to 0 Weakly coupled dimers

Paramagnetic ground state 0, 0iS

2

1

2

1


Excitation: S=1 triplon


2

1



2

1



2

1



2

1



2

1


Energy dispersion away from antiferromagnetic wavevector

2 2 2 2

2x x y y

p

c p c p

spin gap

(exciton, spin collective mode)

TlCuCl3

N. Cavadini, G. Heigold, W. Henggeler, A. Furrer, H.-U. Güdel, K. Krämer and H. Mutka, Phys. Rev. B 63 172414 (2001).

“triplon”

/

For quasi-one-dimensional systems, the triplon linewidth takes

the exact universal value 1.20 at low TBk TBk Te

K. Damle and S. Sachdev, Phys. Rev. B 57, 8307 (1998)

This result is in good agreement with observations in CsNiCl3 (M. Kenzelmann, R. A. Cowley, W. J. L. Buyers, R. Coldea, M. Enderle, and D. F. McMorrow Phys. Rev. B 66, 174412 (2002)) and Y2NiBaO5 (G. Xu, C. Broholm, G. Aeppli, J. F. DiTusa, T.Ito, K. Oka, and H. Takagi, preprint).

Coupled Dimer Antiferromagnet

close to 1 Weakly dimerized square lattice

close to 1 Weakly dimerized square lattice

Excitations: 2 spin waves (magnons)

2 2 2 2p x x y yc p c p


0

spin density wave order parameter: ; 1 on two sublatticesii i

S

S

TlCuCl3

J. Phys. Soc. Jpn 72, 1026 (2003)

1c

Néel state

T=0

Pressure in TlCuCl3

Quantum paramagnet

c = 0.52337(3) M.

Matsumoto, C. Yasuda, S. Todo, and H. Takayama, Phys. Rev. B 65, 014407 (2002)

The method of bond operators (S. Sachdev and R.N. Bhatt, Phys. Rev. B 41, 9323 (1990)) provides a quantitative description of spin excitations in TlCuCl3 across the quantum phase transition (M. Matsumoto, B. Normand, T.M. Rice, and M. Sigrist,

Phys. Rev. Lett. 89, 077203 (2002))

0

0

1c

T=0

in cuprates

Quantum paramagnet

c = 0.52337(3) M.

Matsumoto, C. Yasuda, S. Todo, and H. Takayama, Phys. Rev. B 65, 014407 (2002)

Magnetic order as in La2CuO4 Electrons in charge-localized Cooper pairs

The method of bond operators (S. Sachdev and R.N. Bhatt, Phys. Rev. B 41, 9323 (1990)) provides a quantitative description of spin excitations in TlCuCl3 across the quantum phase transition (M. Matsumoto, B. Normand, T.M. Rice, and M. Sigrist,

Phys. Rev. Lett. 89, 077203 (2002))

0

0

Néel state

22 22 2 22

1 1

2 4!x c

ud xd

cS

LGW theory for quantum criticality

write down an effective action

for the antiferromagnetic order parameter by expanding in powers

of and its spatial and temporal d

Landau-Ginzburg-Wilson theor

erivatives, while preserving

y

:

all s

ymmetries of the microscopic Hamiltonian

S. Chakravarty, B.I. Halperin, and D.R. Nelson, Phys. Rev. B 39, 2344 (1989)

LGW theory for quantum criticality

write down an effective action

for the antiferromagnetic order parameter by expanding in powers

of and its spatial and temporal d

Landau-Ginzburg-Wilson theor

erivatives, while preserving

y

:

all s

ymmetries of the microscopic Hamiltonian

For , oscillations of about 0

constitute the excitationc

triplon

A.V. Chubukov, S. Sachdev, and J.Ye, Phys. Rev. B 49, 11919 (1994)

S. Chakravarty, B.I. Halperin, and D.R. Nelson, Phys. Rev. B 39, 2344 (1989)

22 22 2 22

1 1

2 4!x c

ud xd

cS

B. Mott insulators with spin S=1/2 per unit cell:

Berry phases, bond order, and the breakdown of the LGW paradigm

Mott insulator with two S=1/2 spins per unit cell

Mott insulator with one S=1/2 spin per unit cell


Ground state has Neel order with 0


Destroy Neel order by perturbations which preserve full square lattice symmetry e.g. second-neighbor or ring exchange.The strength of this perturbation is measured by a coupling g.

Small ground state has Neel order with 0

Large paramagnetic ground state with 0

g

g


Destroy Neel order by perturbations which preserve full square lattice symmetry e.g. second-neighbor or ring exchange.The strength of this perturbation is measured by a coupling g.



g

g


Possible large paramagnetic gr Class ound state ( ) with 0A g


ar

ond

c

b

tan j ii

i jij

i S S e

r r

bond bond

Such a state breaks the symmetry of rotations by / 2 about lattice sites,

and has 0, where is the

n bond order parameter

bond



ar

ond

c

b

tan j ii

i jij

i S S e

r r

bond bond




bond



ar

ond

c

b

tan j ii

i jij

i S S e

r r

bond bond




bond



ar

ond

c

b

tan j ii

i jij

i S S e

r r

bond bond




bond



bond bond




ar

ond

c

b

tan j ii

i jij

i S S e

r r

bond



bond bond

Another state breaking the symmetry of rotations by / 2 about lattice sites,

which also has 0, where is the


ar

ond

c

b

tan j ii

i jij

i S S e

r r

bond



bond bond




ar

ond

c

b

tan j ii

i jij

i S S e

r r

bond



bond bond




ar

ond

c

b

tan j ii

i jij

i S S e

r r

bond



bond bond




ar

ond

c

b

tan j ii

i jij

i S S e

r r

bond



bond bond




ar

ond

c

b

tan j ii

i jij

i S S e

r r

bond


Resonating valence bonds

Resonance in benzene leads to a symmetric configuration of valence

bonds (F. Kekulé, L. Pauling)

bond

Different valence bond pairings

resonate with each other, leading

to

Cl

a reso

ass B

nating valen

paramagnet)

ce bond ,

( with 0

liquid

P. Fazekas and P.W. Anderson, Phil Mag 30, 23 (1974); P.W. Anderson 1987

Such states are associated with non-collinear spin correlations, Z2 gauge theory, and topological order.N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991); X. G. Wen, Phys. Rev. B 44, 2664 (1991).

Excitations of the paramagnet with non-zero spin

bond 0; Class A


bond 0; Class A


bond 0; Class A


bond 0; Class A


bond 0; Class A


bond 0; Class A

S=1/2 spinons, , are confined into a S=1

triplon, *~ z z

z


bond 0; Class A


triplon, *~ z z

z

bond 0; Class B


bond 0; Class A


triplon, *~ z z

z

bond 0; Class B


bond 0; Class A


triplon, *~ z z

z

bond 0; Class B


bond 0; Class A


triplon, *~ z z

z S=1/2 spinons can propagate

independently across the lattice

bond 0; Class B

Quantum theory for destruction of Neel order

Ingredient missing from LGW theory: Ingredient missing from LGW theory: Spin Berry PhasesSpin Berry Phases

iSAe

A

iSAe

A





Discretize imaginary time: path integral is over fields on the sites of a cubic lattice of points a



a a

a

Recall = (0,0,1) in classical Neel state;

1 on two

square subl

2

attices ; a aS

a+

0

a

, ,x y



a a

a


1 on two

square subl

2

attices ; a aS

a+

0

a

a a+ 0

oriented area of spherical triangle

formed by and an arbitrary reference , poi, nta hA alf

2 aA

S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)



a a

a


1 on two

square subl

2

attices ; a aS

a+

0

a

a a+ 0






a a

a


1 on two

square subl

2

attices ; a aS

a+

0

a

a a+ 0




a

a



a a

a


1 on two

square subl

2

attices ; a aS

a+

0

a

a a+ 0




a

a



a a

a


1 on two

square subl

2

attices ; a aS

a+

0

a

a a+ 0




a

Change in choice of is like a “gauge transformation”

2 2a a a aA A

0

a



a a

a


1 on two

square subl

2

attices ; a aS

a+

0

a

a a+ 0




a

Change in choice of is like a “gauge transformation”

2 2a a a aA A

0

The area of the triangle is uncertain modulo 4and the action has to be invariant under 2a aA A



exp a aa

i A

Sum of Berry phases of all spins on the square lattice.

,

exp

with "current" of

charges 1 on sublattices

a aa

a

i J A

J

static

,

1a a

ag

2

,

11 expa a a a a a

a aa

Z d i Ag

Partition function on cubic lattice

LGW theory: weights in partition function are those of a classical ferromagnet at a “temperature” g



g

g


2

,

11 expa a a a a a

a aa

Z d i Ag

Partition function on cubic lattice

Modulus of weights in partition function: those of a classical ferromagnet at a “temperature” g

Berry phases lead to large cancellations between different

time histories need an effective action for at large aA g





g

g

2

2 2

,

with

This is compact QED in 3 spacetime dimensions with

static charges 1 on

1exp c

two sublat

tices.

o2

~

sa a a a aaa

Z dA A A i Ae

e g

Simplest large g effective action for the Aa

N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).S. Sachdev and R. Jalabert, Mod. Phys. Lett. B 4, 1043 (1990).

bond (Class A parama

Analysis by a duality mapping shows that this theory

is in a phase with 0

The gauge theory is in a phase (spinons are confined

and only =1 triplons propagate

gnet).always

confining

S

Proliferation of monopoles in the presence of Berry phases.

).


Ordering by quantum fluctuations









g

Phase diagram of S=1/2 square lattice antiferromagnet

T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).

*

Neel order

~ 0z z

(associated with condensation of monopoles in ),A

or

bondBond order 0

1/ 2 spinons confined,

1 triplon excitations

S z

S

A. W. Sandvik, S. Daul, R. R. P. Singh, and D. J. Scalapino, Phys. Rev. Lett. 89, 247201 (2002)

Bond order in a frustrated S=1/2 XY magnet

2 x x y yi j i j i j k l i j k l

ij ijkl

H J S S S S K S S S S S S S S

g=

First large scale (> 8000 spins) numerical study of the destruction of Neel order in a S=1/2 antiferromagnet with full square lattice symmetry

Mott insulators with spin S=1/2 per unit cell:

Berry phases, bond order, and the breakdown of the LGW paradigm

Order parameters/broken symmetry+

Emergent gauge excitations, fractionalization.

C. Cuprate superconductors:Competing orders and recent experiments

Magnetic Mott Insulator

Magnetic Superconductor

Paramagnetic Mott Insulator

Superconductor

BCS , 0 0 =

2 4La CuO

Quantum phasetransitions

High temperaturesuperconductor

BCS , 0 0 =

BCS , 0 0

BCS , 0 0

BCSMinimal LGW phase diagram with and


Magnetic Superconductor


Superconductor

BCS , 0 0 =

2 4La CuO


BCS , 0 0 =

BCS , 0 0

BCS , 0 0

Spin density wave order ,K Spirals……Shraiman, Siggia Stripes……..Zaanen, Kivelson…..

BCSMinimal LGW phase diagram with and

High temperaturesuperconductor



BCS , 0 0 =

2 4La CuO


BCS , 0 0 =



BCS , 0 0 =

2 4La CuO


BCS , 0 0 =

g

Neel order

Bond order

La2CuO4

or

g

Neel order

Bond order

La2CuO4

or

Hole density

Large N limit of a theory with Sp(2N) symmetry: yields

existence of bond order and d-wave superconductivity

S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991); M. Vojta and S. Sachdev, Phys. Rev. Lett. 83, 3916 (1999); M. Vojta, Phys. Rev. B 66, 104505 (2002).

Localized holes

Magnetic, bondand super-conducting

order

g

Neel order

Bond order

La2CuO4

or

Hole density

Large N limit of a theory with Sp(2N) symmetry: yields

existence of bond order and d-wave superconductivity

S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991); M. Vojta and S. Sachdev, Phys. Rev. Lett. 83, 3916 (1999); M. Vojta, Phys. Rev. B 66, 104505 (2002).

Localized holes

J. M. Tranquada, H. Woo, T. G. Perring, H. Goka, G. D. Gu, G. Xu, M. Fujita, and K. Yamada, cond-mat/0401621

Neutron scattering measurements of La15/8Ba1/8CuO4 (Zurich oxide)

Scattering off spin density wave order with

cos

1 1 1 1 1 1 2 , and 2 ,

2 2 8 2 8 2

At higher energies, semiclassical theory predicts

that peaks lead to spin-wave ("light")

i iS N

Q r

Q =

cones.

xy




cos

1 1 1 1 1 1 2 , and 2 ,

2 2 8 2 8 2



i iS N

Q r

Q =

cones.

xy

A. T. Boothroyd, D. Prabhakaran, P. G. Freeman, S.J.S. Lister, M. Enderle, A. Hiess, and J. Kulda, Phys. Rev. B 67, 100407 (2003).

La5/3Sr1/3NiO4

1 1 1 1Ni has 1; 2 ,

2 6 2 6S

Q




cos

1 1 1 1 1 1 2 , and 2 ,

2 2 8 2 8 2



i iS N

Q r

Q =

cones.

xy


La5/3Sr1/3NiO4

Spin waves: J=15 meV, J’=7.5meV



cos

1 1 1 1 1 1 2 , and 2 ,

2 2 8 2 8 2



i iS N

Q r

Q =

cones.

xy


La5/3Sr1/3NiO4

Spin waves: J=15 meV, J’=7.5meV



Observations in La15/8Ba1/8CuO4

are very different and do not obey spin-wave model.

Similar spectra are seen in most hole-doped cuprates.

xy

“Resonance peak”


Red lines: triplon excitation of a 2 leg

ladder with exchange J=100 meV







J. M. Tranquada et al., cond-mat/0401621

xy

Spectrum of a two-leg ladder

Possible simple microscopic model of bond order

Spin waves

Triplons

“Resonance peak”

• M. Vojta and T. Ulbricht, cond-mat/0402377• G.S. Uhrig, K.P. Schmidt, and M. Grüninger, cond-mat/0402659• M. Vojta and S. Sachdev, unpublished.


xy

Bond operator (S. Sachdev and R.N. Bhatt, Phys. Rev. B 41, 9323 (1990)) theory of coupled-ladder model,

M. Vojta and T. Ulbricht, cond-mat/0402377


xy

Numerical study of coupled ladder model, G.S. Uhrig, K.P. Schmidt, and M. Grüninger,

cond-mat/0402659


xy

LGW theory of magnetic criticality in the presence of static bond order, M. Vojta and S. Sachdev, to appear.

G.S. Uhrig, K.P. Schmidt, and M. Grüninger, cond-mat/0402659

Conclusions

I. Theory of quantum phase transitions between magnetically ordered and paramagnetic states of Mott insulators:

A. Dimerized Mott insulators: Landau-Ginzburg-Wilson theory of fluctuating magnetic order parameter.

B. S=1/2 square lattice: Berry phases induce bond order, and LGW theory breaks down. Critical theory is expressed in terms of emergent fractionalized modes, and the order parameters are secondary.

Conclusions

I. Theory of quantum phase transitions between magnetically ordered and paramagnetic states of Mott insulators:

A. Dimerized Mott insulators: Landau-Ginzburg-Wilson theory of fluctuating magnetic order parameter.

B. S=1/2 square lattice: Berry phases induce bond order, and LGW theory breaks down. Critical theory is expressed in terms of emergent fractionalized modes, and the order parameters are secondary.

Conclusions

II. Competing spin-density-wave/bond/superconducting orders in the hole-doped cuprates.

• Main features of spectrum of excitations in LBCO modeled by LGW theory of quantum critical fluctuations in the presence of static bond order across a wide energy range.

• Predicted magnetic field dependence of spin-density-wave order observed by neutron scattering in LSCO. E. Demler, S. Sachdev, and Y. Zhang, Phys.Rev. Lett. 87, 067202 (2001); B. Lake et al. Nature, 415, 299 (2002); B. Khaykhovich et al. Phys. Rev. B 66, 014528 (2002).

• Predicted pinned bond order in vortex halo consistent with STM observations in BSCCO. K. Park and S. Sachdev Phys. Rev. B 64, 184510 (2001); Y. Zhang, E. Demler and S. Sachdev, Phys. Rev. B 66, 094501 (2002); J.E. Hoffman et al. Science 295, 466 (2002).

• Energy dependence of LDOS modulations in BSCCO best modeled by modulations in bond variables. M. Vojta, Phys. Rev. B 66, 104505 (2002); D.

Podolsky, E. Demler, K. Damle, and B.I. Halperin, Phys. Rev. B 67, 094514 (2003); C. Howald, H. Eisaki, N. Kaneko, and A. Kapitulnik, Phys. Rev. B 67, 014533 (2003).

Conclusions

II. Competing spin-density-wave/bond/superconducting orders in the hole-doped cuprates.

• Main features of spectrum of excitations in LBCO modeled by LGW theory of quantum critical fluctuations in the presence of static bond order across a wide energy range.

• Predicted magnetic field dependence of spin-density-wave order observed by neutron scattering in LSCO. E. Demler, S. Sachdev, and Y. Zhang, Phys.Rev. Lett. 87, 067202 (2001); B. Lake et al. Nature, 415, 299 (2002); B. Khaykhovich et al. Phys. Rev. B 66, 014528 (2002).

• Predicted pinned bond order in vortex halo consistent with STM observations in BSCCO. K. Park and S. Sachdev Phys. Rev. B 64, 184510 (2001); Y. Zhang, E. Demler and S. Sachdev, Phys. Rev. B 66, 094501 (2002); J.E. Hoffman et al. Science 295, 466 (2002).

• Energy dependence of LDOS modulations in BSCCO best modeled by modulations in bond variables. M. Vojta, Phys. Rev. B 66, 104505 (2002); D.

Podolsky, E. Demler, K. Damle, and B.I. Halperin, Phys. Rev. B 67, 094514 (2003); C. Howald, H. Eisaki, N. Kaneko, and A. Kapitulnik, Phys. Rev. B 67, 014533 (2003).

Conclusions

III. Breakdown of LGW theory of quantum phase transitions with magnetic/bond/superconducting orders in doped Mott insulators ?

Conclusions

III. Breakdown of LGW theory of quantum phase transitions with magnetic/bond/superconducting orders in doped Mott insulators ?

Documents

Colloquium article in Reviews of Modern Physics 75 , 913 (2003)